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Nonuniform exponential contractions and Lyapunov sequences. (English) Zbl 1173.37029
The paper deals with a nonautonomous dynamics with discrete time obtained from the product of linear operators. For this dynamics, it is shown that a nonuniform exponential contraction can be completely characterized in terms of so-called strict Lyapunov sequences. Nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. The paper also presents “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. It is also shown how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. In addition, an appropriate version of the results obtained is described in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.
MSC:
37D25Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
93D30Scalar and vector Lyapunov functions
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